This particular myth is, strictly speaking, only a semi-myth, but I will debunk it anyway. The fact is that modern firearms used with modern ammunition need not be cleaned after every use.

The origins of this belief are two: first, from shooting black powder; and second, from military training and policy.

Black powder (and most of the so-called "replica" powders like Pyrodex and 777) leave residue in the barrel and cylinder that are indeed corrosive. These residues are hygroscopic and if a gun used with black powder isn't cleaned promptly, will cause corrosion and pitting, detrimental to accuracy and function. So in fact, black powder firearms should be cleaned after every use. Just what is used to clean them isn't terribly important, assuming it's a product designed for such use, but it should be done. Personally I always clean with plain old soap and hot water. The residue is water soluble, and a flush followed by use of soap will get it out. There are numerous proprietary cleaners on the market that are alleged to work as well, but soap and water don't cost anything.

In the late 19th Century a sort of transitional propellant, sold under the name "Lesmok," was available. Lesmok, sometimes called "semi-smokeless" powder, was a mixture of black powder and guncotton, used until the early 1940's. Ammunition using it was intended for indoor target shooting (a quaint and now mostly vanished pastime), hence the name. Lesmok ammunition often carried a label like the one shown here from a box of Winchester .22 Shorts. By the time this ammunition was made (in 1929) the company had long since abandoned the old corrosive priming but the label does warn that Lesmok does leave "fouling" in the bore; to avoid it completely, the user is advised to shoot smokeless powder.

Modern powders are very gentle compared to black; coupled with non-corrosive priming the barrel need not be cleaned until for some reason accuracy falls off. It is actually the case that the more a .22 rifle is shot with modern ammunition, the less likely it is to be damaged: .22 ammunition is "externally lubricated" with a waxy compound that will deposit a coat in the bore that protects it from harm. If you ever encounter a .22 with a corroded bore, it's a safe bet it predates 1920 or so and was used with black powder cartridges. I once knew someone who was a top-level smallbore competitor on the Virginia state rifle team. He was horrified by the idea of cleaning the barrels of his target guns, arguing (quite correctly) that running a bristle brush down the barrel did more harm than the bullets did.

So to that extent, the myth isn't a myth: but modern powders and priming are far, far gentler than black powder (let alone the old corrosive priming mixes used in the 19th and early 20th Centuries, which incorporated mercuric and chlorate compounds).

Now, on to the military origin of this myth.

Anyone who has served in a military service knows that the phrase, "The Devil finds work for idle hands," is taken to heart by drill sergeants and unit commanders. Keeping the enlisted ranks busy is something they believe is necessary. Hence soldiers are ordered to clean their firearms every time they're used. Not because the guns need it, but because it's something that keeps the ranks busy.

If there is more damage done by cleaning than by shooting, why then do armies continue to insist on it being done every time a gun is shot? Again, it hearkens back to the days when armies used black powder, and regular cleaning was indeed absolutely necessary. The tradition has carried on down the long, long line of military training to this day.

And...keep in mind that military organizations don't have to worry too much about firearms being worn out. When an M16 is finally shot out—something that rarely happens to civilian firearms—the unit commander simply indents the supply chain for a new one. Private citizens have to pay out of pocket to replace a gun, soldiers don't.